On the distribution of Weierstrass points on singular curves (Q1092967)

Let X denote an integral, projective Gorenstein curve over \({\mathbb{C}}\) and \({\mathcal L}\) an invertible sheaf on X. A definition of a Weierstraß point P of \({\mathcal L}\) is given. If P is a smooth point of X, P is a Weierstraß point of \({\mathcal L}\) of order n if and only if \(h^ 0({\mathcal L}^ n(-h^ 0({\mathcal L}^ n)\cdot P))>0\). Hence the definition generalizes the classical definition for smooth curves. On the other hand any singular point of X is a Weierstraß point. Whereas for a smooth curve the set of Weierstraß points of X (of some order) is dense in X [\textit{B. A. Olsen}, Ann. Math., II. Ser. 95, 357-364 (1972; Zbl 0232.14017)] an example of a rational nodal curve X and an invertible sheaf \({\mathcal L}\) on X is given, for which this is not the case. The explanation of this is that in a family of curves the Weierstraß points tend towards the singularities.